Number
For other uses, see Number (disambiguation). A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, number may refer to a symbol, a word, or a mathematical abstraction. In mathematics, the notion of number has been extended over the centuries to include 0, negative numbers, rational numbers such as 1/2 and -2/3, real numbers such as √2 and π, and complex numbers, which extend the real numbers by adding a square root of −1. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic. The same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. For example, in Western society the number 13 is regarded as unlucky, and "a million" may signify "a lot." Though it is now regarded as pseudoscience, numerology, the belief in a mystical significance of numbers, permeated ancient and medieval thought. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Today, number systems are considered important special examples of much more general categories such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance. Numberals Main article: Numeral system Numbers should be distinguished from numerals, the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system. Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit "5" or by the Roman numeral "Ⅴ". Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which have many advantages, such as representing large numbers with only a few symbols. The Roman numerals require extra symbols for larger numbers. Main classification See also: List of types of numbers Different types of numbers have many different uses. Numbers can be classified into sets, called number '''systems, such as the natural numbers and the real numbers. The same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals. '''Main number systems Each of these number systems may be considered as a proper subset of the next one. This means that each one is canonically isomorphic to a proper subset of the next one, and that there is generally no problem with the abuse of notation consisting of identifying each number system with a subset of the next one. This is expressed, symbolically, by writting : N '''belong to Z belong to '''Q '''belong to '''R '''belong to C''' Subclasses of the integers Even and odd numbers Main article: Even and odd numbers An even number is an integer that is "evenly divisible" by two, that is divisible by two without remainder; an odd number is an integer that is not even. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) Equivalently, another way of defining an odd number is that it is an integer of the form n'' = 2''k + 1, where k'' is an integer, and an even number has the form ''n = 2''k'' where k'' is an integer. '''Prime numbers' Main article: Prime number A prime number is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. The prime numbers have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. The study of these questions belong to number theory. An example of a question that is still unanswered is whether every even number is the sum of two primes. This is called Goldbach's conjecture. A question that has been answered is whether every integer greater than one is a product of primes in only one way, except for a rearrangement of the primes. This is called fundamental theorem of arithmetic. A proof appears in Euclid's Elements. Other classes of integers Many subsets of the natural numbers have been the subject of specific studies and have been named, often after the first mathematician that has studied them. Example of such sets of integers are Fibonacci numbers and perfect numbers. For more examples, see Integer sequence. Subclasses of the complex numbers Algebraic, irrational and transcendental numbers Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. Real numbers that are not rational numbers are called irrational numbers. Complex numbers which are not algebraic are called transcendental numbers. The algebraic numbers that are solutions of a monic polynomial equation with integer coefficients are called algebraic integers. Computable numbers A computable number, also known as recursive number, is a real number such that there exists an algorithm which, given a positive number N as input, produces the first ''N ''digits of the computable number's decimal representation. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus. The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers. The computable numbers may be viewed as the real numbers that may be exactly represented in a computer: a computable number is exactly represented by its first digits and a program for computing further digits. However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not. The set of computable numbers has the same cardinality as the natural numbers. Therefore, almost all real numbers are non-computable. However, it is very difficult to produce explicitly a real number that is not computable. Extensions of the concept p-adic numbers Hypercomplex numbers Transfinite numbers Nonstandard numbers Thể_loại:MATH